The R2 value is coming out to almost close to 0.999 both pseudo first and second order kinetics, for all the adsorbate concentration values. Yet the value of adsorption capacity at equilibrium (qe) is better predicted by pseudo first order kinetics.
Dear Shantanu Banerjee,
If your kinetic model best fits Pseudo first order reaction plot by by giving r2 value close to 1, it indicates that the reaction is more inclined towards physisorption. Similarly if the reaction fits well to Pseudo second order model it indicates an inclination towards chemisorption. Several reactions in general follow chemisorption initially and that too over a very short period of time
Thank you all for your answers. I can conclude that error functions (SSE, RMSE) can give me a better understanding of the kinetics.
Hello Shantanu,
Motulsky & Christopoulos 2003 explained that before nonlinear regression was readily available, shortcuts were developed to analyze nonlinear data. The idea was to transform the data to create a linear graph, and then analyze the transformed data with linear regression. The problem with these methods is that they cause some assumptions of linear regression to be violated. Since the assumptions of linear regression are violated, the values derived from the slope and intercept of the regression line are not the most accurate determinations of the parameters in the model.
Mayers 1990 also cautioned that such transformations implicitly alter the error structure and may also violate the error variance and normality assumptions of the standard least square method.
Consequently, I suggest that you use the linear parameters you obtained from the linear regression and generate the theoretical non-linear curve. Then compare the theoretical curve to your experimental data. You can assess the goodness of fit using the RPE, non-linear R2, SSE or some of the other error functions mentioned by our colleagues. Or you can simply conduct non-linear regression to help elucidate which model fits your data best.
However, several authors have iterated that many models were originally developed based on gas adsorption and other theoretical concepts, thus their applicability may be only limited to the mathematical representation of your data and mechanistic inferences should be drawn carefully.
Good Luck!
Nothing surprising. Both functions are increasing, they have an upper limit and do not have inflection points. Of course, the exponential function (PFO) is faster increasing than the hyperbolic function (PSO). But just reduce the value of k in the exponential function and you can get a similar R2 value. That the whole secret. Regards,
Of course, R2 is not good measure of fitting. In addition, it is better to use average mean error and compare directly qtexp and qtcal. Regards,
If the adsorption kinetics is following both pseudo first and second order kinetics, it means that the adsorption process is complex and involves more than one rate-limiting step. The pseudo first order kinetics assumes that the adsorption rate is proportional to the number of vacant sites on the adsorbent surface, while the pseudo second order kinetics assumes that the adsorption rate is proportional to the square of the number of vacant sites. Both models can fit the experimental data well, but they may not reflect the true mechanism of adsorption. Therefore, it is important to compare the estimated parameters, such as qe and k, with the experimental values and other criteria, such as error analysis and initial adsorption rate, to select the best model for the adsorption system12.
In your case, since the value of qe is better predicted by pseudo first order kinetics, it suggests that this model is more suitable for your system. However, you should also check the values of k and the initial adsorption rate to confirm this. The value of k should be consistent and independent of the initial concentration, and the initial adsorption rate should agree with the experimental data3. You should also calculate the error functions, such as sum of squares error (SSE), hybrid fractional error function (HYBRID), and Marquardt’s percent standard deviation (MPSD), to evaluate the accuracy and precision of the models4. The lower the error values, the better the model fit.
Some references that explain these concepts in more detail are:
- Kinetics of solute adsorption at solid/solution interfaces: a theoretical development of the empirical pseudo-first and pseudo-second order kinetic rate equations, based on applying the statistical rate theory of interfacial transport by W. Rudzinski and W. Plazinski, Adsorption, vol. 15, no. 3, pp. 181-192, 2009.
- Pseudo-second-order kinetic equations for modeling adsorption systems for removal of lead ions using multi-walled carbon nanotube by Y. Ho and G. McKay, Applied Mechanics and Materials, vol. 625, pp. 15-18, 2014.
- Comparison of linear and non-linear regression analysis to determine pile bearing capacity by A. Sengoz and B. Gulluoglu, Journal of Applied Sciences, vol. 7, no. 2, pp. 258-263, 2007.
- Error analysis in using pseudo-second-order model for liquid-phase adsorption: A review by S.-Y. Foo and B.-H. Hameed, Journal of Hazardous Materials, vol. 244-245, pp. 277-288, 2013.
Chapter Modelling of Adsorption Kinetic Processes—Errors, Theory and...
Article Modeling of sorption kinetics: The pseudo-second order equat...
https://iosrjournals.org/iosr-jac/papers/vol12-issue2/Series-1/B1202011116.pdf
Pseudo-second-order kinetic equations for modeling adsorption...
https://www.intechopen.com/chapters/63161
https://link.springer.com/article/10.1007/s10450-013-9529-0
https://iosrjournals.org/iosr-jac/papers/vol12-issue2/Series-1/B1202011116.pdf
Pseudo-second-order kinetic equations for modeling adsorption...
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